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The authors present a set of statistical decision rules based on linear regression models that can be implemented in an automated trend system to assist stability studies.
Submitted: Feb. 22, 2016.
Accepted: Apr. 7, 2016.
The authors present a set of statistical decision rules based on linear regression models that can be implemented in an automated trend system to assist stability studies. The models combine historical stability and analytical method data with data from stability studies, and allow the responsible person to routinely evaluate stability results based on statistical tools, without the need for expert statistical assistance. The system provides a fast and standardized framework for evaluating parameters that approximately follow a linear degradation path or are constant.
Evaluation of data from stability studies is a central part of the control strategy of pharmaceutical products and is a GMP requirement (1). The purpose is to ensure the safety and efficacy of the product by confirming that the stability is as expected and that it will continue to meet quality specifications until expiry. Stability studies can be part of the development program for new products or the ongoing stability program for marketed products. The studies are typically conducted both at long-term storage conditions and at accelerated conditions.
For stability studies on marketed products, the objective is to confirm that the stability profile follows the trend of earlier batches. Unexpected results may either indicate that the batch is out-of-trend or that the result is out-of-trend (OOT). A typical approach to evaluate the data is to consider the following three questions (2):
The evaluation can be performed subjectively by an analyst, but it requires long experience with the analytical method and the product and its distinct properties. Also, different analysts or different laboratories may conduct the trending; they may have different experience and evaluate the data differently as a result. However, an objective evaluation requires data from different sources to be combined, namely the precision of the analytical method and the stability trend of historical batches and their associated uncertainty, and this is a burden both practically and statistically.
Statistical tools can control the risks of false alarms, when the product and result are actually within the expected range, and the risk of overlooking an OOT. The factors of uncertainty that need to be considered are:
Unless a system is in place that facilitates the combination and statistical evaluation of data in an automated and standardized manner, the evaluation of stability data will be laborious and may require expert statistical assistance, which is usually not readily available at all the facilities where data are generated and evaluated.
A number of different approaches for evaluating stability data from a statistical perspective have been proposed in recent years (2-5). In this paper, the authors consider only parameters that follow a linear stability trend (or are constant). In this approach, the analysis is based on linear regression models that combine the efficiency of a parametric statistical model with the practical aspect of being relatively simple and intuitive.
From the authors’ experience, the vast majority of parameters that are followed in stability are approximated well by zero or first-order kinetic reactions, which lend themselves to linear regression analyses. Parameters that do not develop linearly must be evaluated, for instance, by tolerance interval methods by time point (3), or by more advanced kinetic models of the stability profile. These methods will not be considered here.
An overview of the system is provided in the following sections. Statistical details are deferred to the appendix.
The system is illustrated in Figure 1. The system supports a work flow where the stability responsible person routinely evaluates and releases results in a stability study as they are available. Stability data is stored in a laboratory information management system (LIMS). To evaluate the trend questions discussed in the previous section, historical data and data on the analytical variability of the method are needed. These data are stored in a database with tables for each product.
The combination of the two data sources and the statistical analysis and presentation of results is implemented in a computer program (JMP, SAS Institute) (6), but other systems for data analysis and visualization can be used. The evaluation of results and alerts is conducted on a computer screen.
The parameter table with historical data
Historical stability data is summarized in a parameter table (see Table I) for each product. The table should be based on batches and results that are representative of the current product and analytical methods.
The parameter table should be established based on statistical analysis of historical stability data that are representative of the current product. For new products, typically data from the new drug application (NDA) stability studies and other development stability studies will be used. For marketed products, the body of historical routine stability data can be used.
The analysis of the historical data should be based on a regression analysis, in which the average stability trend is determined. In the model, each batch should have its own intercept to account for batch-to-batch variation in the starting level. If the stability slope varies slightly from batch-to-batch due to random variations, for instance in raw materials or input factors, a mixed model with random slopes can be used (5).
The intermediate precision of the analytical method should preferably be estimated as the residual variation in historical stability data, because this estimate will cover long-term variation in the method and also any other variation in stability studies, for instance, due to sampling and handling of the samples. Alternatively, method validation data or variation in control samples can be used.
The construction of the parameter table is typically a large task and may require a cross-functional team of analytical chemists, product responsible chemists, and statisticians. It is advisable to ensure careful documentation and control of the parameter table because it is the cornerstone of the stability trend evaluation.
Generally, the parameter table need only be established once for each product, but it may be necessary to update the table over time if there are changes to the stability profile of the product or to the analytical methods, or if the initial parameter table is based on a relatively small body of stability data and more precise estimates are obtained over time.
The parameter table summarizes all the historical knowledge of the product and the analytical methods in a single table. Thus, there is a wealth of information in the table, and the creation of the table ensures that the expectation of the stability study is clear across the organization. By using the same parameter table for trending, consistency in the evaluation of the data across persons, departments, and sites is ensured, which is an important benefit of the system.
When conducting routine trending, stability data are retrieved from the LIMS and combined with the parameter table. The system processes the data and presents a graph for each parameter, batch, and storage condition. The graphs illustrate the data and summarize the statistical evaluation of the three trend questions.
Is the latest result comparable with the results previously seen for the same batch in the study?
This trend is evaluated by a prediction interval based on the stability results for each batch, excluding the latest result. If the latest result falls within the prediction interval, it can be concluded that it follows the trend seen so far, within the expected uncertainty range.
Typically, a 99% prediction interval will be used to have a reasonably low risk (1%) of a false alarm. This interval corresponds approximately to ±3 standard deviations around the expected value.
The historical stability slope in the parameter table is not used in this evaluation, but the historical intermediate precision of the method is used to calculate the variance of the result.
The result of the analysis is indicated graphically by plotting the data with the regression line, calculated with the latest result excluded, and by overlaying ±3 standard deviation error bars on the latest result. This approach provides a simple visual check for whether the result is within the expected range. The conclusion of the statistical analysis is illustrated visually by plotting the latest result with a red symbol, if the result is outside the 99% prediction interval. An example is provided in Figure 2.
Is the development of the parameter comparable to the development of the same parameter in historical studies?
This trend is addressed by a regression analysis, in which the estimated slope of the current batch is compared with the expected slope from the parameter table. Based on a t-test, the statistical significance of any difference can be assessed, accounting for the uncertainty of both the current estimated slope and the expected slope. The uncertainty of the expected slope can express both estimation uncertainty and, if relevant, random batch-to-batch variation in the slope (5). Typically, a significance level of 1% will be used to avoid too many false alarms, corresponding to the 99% intervals used above.
The result of the analysis is indicated graphically by plotting the regression line for the batch (the green line in Figure 3) as well as a line with the expected slope (dotted line in Figure 3). If a statistically significant difference is observed, all points can be plotted with a separate color to provide the stability responsible person with a clear visual indication that this statistically significant difference needs to be evaluated and possibly investigated further.
Can compliance with the specification limits be expected to be maintained until the end of study?
This analysis is conducted following the principles in (7) by evaluating if the 95% confidence interval for the batch intersects the specification limit before the end of shelf life. A one- or two-sided confidence interval is used depending on whether the specification is one- or two-sided, respectively.
If the batch is confirmed to be OOT and there is less than 95% confidence that it will comply with the specification during shelf life, a compliance alert is raised (see Figure 4). The evaluation of criticality is not only a statistical exercise, but the statistical result may be used to evaluate the effect of reducing shelf life or other mitigations.
In the practical use of the system, all data for a given time point are evaluated and a graphical overview of the different parameters, batches, and storage conditions presented. The graphical illustrations of alerts make it easy to get an overview of the data. In case one or more alerts are identified, summary tables with estimates and statistical details are available to interpret the findings.
When evaluating alerts, the trend responsible person should be aware of a number of pitfalls and understand the limitations of the methods used:
The computer system should be validated for GMP-use. However, by building the system on existing validated computer systems, the validation effort is relatively smaller than if the system was built from scratch.
Comparison with other methods
As discussed, the methods presented rely on linear trend models with normally distributed errors. They are, therefore, less general than OOT methods that do not rely on these assumptions, such as the ”change-from-previous” type methods and by-time-point methods presented in references 2 and 3, but they provide a simpler and more efficient setup when the assumptions are fulfilled.
The methods can be compared with other published regression methods as follows:
The trend analysis system provides the trend responsible person with exact and reproducible results for evaluating stability data. It makes the evaluation of data objective and standardized, and provides greater flexibility in terms of who does the trending.
The system provides valuable summary measures for each batch, such as the estimated slope with confidence limits, a statistical test for whether the batch is comparable with historical batches, and the expected shelf life based on extrapolation of confidence intervals. The system makes it easy to account for the different sources of uncertainty in the evaluation of the data and thus provides control over the risk of false alarms and the risk of overlooking an OOT.
The system is relatively simple to implement, validate, and maintain, and can be based on a statistical software package such as JMP and existing database solutions, such as LIMS. The statistical methods strike a reasonable compromise between being relatively simple, based on linear regression model for each batch, yet sufficiently complex to handle, for instance, mixed effect models with random variation in the slope between batches.
Generating the database of parameter tables for all products requires analyses of historical data. Though this effort is a prerequisite for conducting a trend analysis, whether a trend system is used or not, the practical work of establishing, documenting, and maintaining the parameter tables in a system such as this should not be underestimated.
The system is not designed to encompass all parameters, and some level of “manual” trending should, therefore, be expected even with this system. Parameters that do not follow a linear pattern or ordinal responses cannot be analyzed by the system currently. Also, impurity data that are truncated below limit of quantification may need to be trended by other methods. One could extend the system, for instance, by including functionality for transforming responses to linearize the trend, or to include tolerance intervals methods. Still, it is important that the results of the analyses are intuitive and easy to interpret, and this feature should be a cardinal point when extending the system.
Niels Væver Hartvig, PhD, is a principal specialist, email@example.com, tel.: +45 30790913, and Liselotte Kamper is a chemist, both at Novo Nordisk A/S, Smørmosevej 17-19, DK-2880 Bagsværd, Denmark.
The system has been developed by a team of stability responsible persons in Novo Nordisk. In particular, the authors acknowledge valuable discussions and input from Marika Ejby Reinau, Jens Krogh Rasmussen, Helle Lindgaard Madsen, Lone Steenholt, Carsten Berth, and Karin Bilde.
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Vol. 41, No. 1
When referring to this article, please cite it as N. Hartvig and L. Kamper, “A Statistical Decision System for Out-of-Trend Evaluation," Pharmaceutical Technology 41 (1) 2017.